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  • Holonomies for connections with values in L∞-algebras

    Camilo Arias Abad∗ and Florian Schätz†

    April 28, 2014

    Abstract

    Given a flat connection α on a manifold M with values in a filtered L∞-algebra g, we construct a morphism hol∞α : C•(M) → BÛ∞(g), generalizing the holonomies of flat connections with values in Lie algebras. The construction is based on Gugenheim’s A∞- version of de Rham’s theorem, which in turn is based on Chen’s iterated integrals. Finally, we discuss examples related to the geometry of configuration spaces of points in Euclidean space Rd, and to generalizations of the holonomy representations of braid groups.

    Contents

    1 Introduction 2

    2 The universal enveloping algebra 4 2.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 The enveloping algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

    3 Complete L∞-algebras 9 3.1 Generalities about complete L∞-algebras . . . . . . . . . . . . . . . . . . . . . . 9 3.2 Compatibility with various functors . . . . . . . . . . . . . . . . . . . . . . . . . . 10

    4 Parallel transport 14 4.1 A∞ de Rham Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4.2 Holonomies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

    5 Flat connections on configuration spaces 19 5.1 Flat connections and rational homotopy theory . . . . . . . . . . . . . . . . . . . 19 5.2 Flat connections on configuration spaces . . . . . . . . . . . . . . . . . . . . . . . 20

    5.2.1 Kontsevich’s models for configuration spaces . . . . . . . . . . . . . . . . 20 5.2.2 The Ševera–Willwacher connections . . . . . . . . . . . . . . . . . . . . . 22

    5.3 Drinfeld–Kohno construction in higher dimensions . . . . . . . . . . . . . . . . . 23

    A The completed bar complex 27

    ∗Max Planck Institut für Mathematik, Bonn. camiloariasabad@gmail.com. Partially supported by SNF Grant 200020-131813/1 and a Humboldt Fellowship. †Centre for Quantum Geometry of Moduli Spaces, Ny Munkegade 118, DK-8000 Aarhus C, Denmark. flo-

    rian.schaetz@gmail.com. Supported by ERC Starting Grant no. 279729 and the Danish National Research Foundation grant DNRF95 (Centre for Quantum Geometry of Moduli Spaces - QGM).

    1

  • 1 Introduction

    In this note we propose an answer to the following question: Assume that M is a smooth manifold, g an L∞-algebra and α a flat connection on M with values in g, i.e., a Maurer–Cartan element of the L∞-algebra g⊗̂Ω(M); what are the holonomies associated to the flat connection α? Our answer differs from those that have appeared in the literature, such as [24, 26, 27, 32, 34], where various notions of two-dimensional parallel transport are considered. In order to motivate our answer, let us first discuss the case where g = EndV is the Lie algebra of endomorphisms of a finite-dimensional vector space. In this case, α is just a flat connection on the trivial vector bundle V , and by solving the differential equation for parallel transport, one obtains the holonomy ĥol(σ) ∈ EndV ⊂ U(EndV ) associated to a path σ : I → M . One can view this whole assignment as an element ĥol of EndV ⊗C•(M), the differential graded algebra of EndV - valued smooth singular cochains on M . The flatness of α implies the homotopy invariance of the holonomy. This corresponds to the fact that ĥol is a Maurer–Cartan element. Indeed, an element β ∈ EndV ⊗ C1(M) is a Maurer–Cartan element precisely if it is homotopy invariant in the sense that for any two-dimensional simplex one has

    − = 0.

    Here the bold edges represent holonomies associated to the corresponding paths, and concatena- tion of paths corresponds to multiplication in the algebra EndV . Observe that a Maurer–Cartan element of EndV ⊗C•(M) corresponds naturally to a morphism of differential graded coalgebras C•(M)→ B(EndV ).

    Using the explicit iterated integral formulas for the parallel transport, one can show that this morphism factors through the bar coalgebra of the (completed) universal enveloping algebra of EndV :

    C•(M) holα //

    &&

    BÛ(EndV )

    p

    �� B(EndV ).

    This construction works for any filtered Lie algebra g, and we conclude that the holonomies of a flat connection with values in g can be interpreted as a morphism of differential graded coalgebras holα : C•(M)→ BÛ(g), where BÛ(g) denotes the bar construction of the completion of the universal enveloping algebra U(g).

    The case where the L∞-algebra g is the graded Lie algebra of endomorphisms of a graded vector space V corresponds to holonomies of flat Z-graded connections. This has been studied recently by Igusa [16], Block and Smith [8], and Arias Abad and Schätz [3], and ultimately relies on Gugenheim’s [13] A∞-version of de Rham’s theorem. In turn, Gugenheim’s construction is based on Chen’s theory of iterated integrals [9]. We extend this approach to flat connections with values in L∞-algebras. The holonomy of α is a morphism of differential graded coalgebras hol∞α : C•(M)→ BÛ∞(g).1

    1Throughout the introduction, we gloss over the technical issue that one has to work with the completed bar complex B̂Û∞(g) of Û∞(g), which is not a differential graded coalgebra, because its “comultiplication” does not map into the tensor product, but into the completion. See Appendix A for details.

    2

  • We first need to explain what the universal enveloping algebra U∞(g) of an L∞-algebra g is. Several proposals for a definition of the enveloping algebra of an L∞-algebra exist in the literature, e.g., [2, 6, 20]. Following [6], we use the idea of defining the enveloping algebra via the strictification S(g) of the L∞-algebra g. The differential graded Lie algebra S(g) is naturally quasi-isomomorhic to g, and we define the enveloping algebra of g to be that of its strictification. Our main result is as follows:

    Theorem 4.11. Suppose that α is a flat connection on M with values in a filtered L∞-algebra g. Then there is a natural homomorphism of differential graded coalgebras

    hol∞α : C•(M)→ BÛ∞(g).

    In order for this notion of holonomy to be reasonable, it should be consistent with the standard definition in the case of Lie algebras. Indeed, in the case where g is a Lie algebra, the usual parallel transport provides a holonomy map:

    hol : C•(M)→ BÛ(g).

    On the other hand, there is a natural map of differential graded coalgebras

    BÛ(ρ) : BÛ∞(g)→ BÛ(g),

    and the following diagram commutes:

    C•(M) hol∞α //

    holα $$

    BÛ∞(g)

    BÛ(ρ) ��

    BÛ(g).

    The notion of holonomy on which Theorem 4.11 is based admits a rather visual description. Given any filtered differential graded algebra (A, ∂), a morphism of differential graded coalgebras φ : C•(M)→ BÂ corresponds to a Maurer–Cartan element φ in the algebra A⊗̂C•(M), which is an element in the vector space Hom(C•(M), A). Thus, φ can be interpreted as a rule that assigns to each simplex in M an element of the algebra Â, which we think of as being the holonomy associated to that simplex.

    Since the algebra A⊗̂C•(M) is bigraded, the condition for φ to be Maurer–Cartan decomposes into a sequence of equations. In degree 0, the condition is that φ assigns to every point p ∈M a Maurer–Cartan element of Â. This implies that if we set ∂p := ∂+[φ(p), ], then ∂p ◦∂p = 0. Let us denote the complex (A, ∂p) by Ap. Given a simplex σ : ∆k →M , we denote the commutator between the operation of multiplying by φ(σ) and of applying the differentials associated to the first and last vertex of σ by [∂, φ(σ)], i.e., [∂, φ(σ)] := ∂vk ◦ φ(σ)− (−1)1+|σ|φ(σ) ◦ ∂v0 .

    The Maurer–Cartan equation in degree 1 is

    ∂, = 0,

    which says that multiplication by the holonomy associated to a path is an isomorphism between the complexes Av0 and Av1 . The equation in degree 2 reads

    3

  • ∂, = − ,

    requiring that the two isomorphisms between the complexes Av0 and Av2 are homotopic, with a specified homotopy given by the holonomy associated to the triangle.

    Similarly, for the tetrahedron one obtains

    ∂, =

    − + − .

    Our main motivation to develop this version of parallel transport is the appearance of certain flat connections on configuration spaces Confd(n) of n points in Euclidean space Rd. In dimension d = 2 these connections were introduced and studied by Ševera and Willwacher in [28]. There, the flat connections mentioned above yield a homotopy between the formality maps for the little disks operad of Kontsevich [19] and Tamarkin [31], respectively, provided that in the second one the Alekseev–Torossian associator is used.

    In Section 5, we discuss these connections on configuration spaces. We first explain a link be- tween rational homotopy theory and the theory of flat connections with values in L∞-algebras. We then describe Kontsevich’s model ∗Graphsd(n) of Confd(n) and the corresponding flat con- nections SWd(n), extending the construction of Ševera and Willwacher to higher dimensions. Finally, we demonstrate how to use this machinery to construct actions of the ∞-groupoid of Confd(n) on representations of quadratic differential graded Lie algebras, generalizing the holonomy representations of the braid groups.

    Acknowledgements

    We thank Alberto Cattaneo, Yaël Frégier, Pavol Ševera, and Tho